Graded identities of matrix algebras and the universal graded algebra
Authors:
Eli Aljadeff, Darrell Haile and Michael Natapov
Journal:
Trans. Amer. Math. Soc. 362 (2010), 31253147
MSC (2000):
Primary 16W50, 16R10, 16R50; Secondary 16S35, 16K20
Published electronically:
January 7, 2010
MathSciNet review:
2592949
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Abstract: We consider fine group gradings on the algebra of by matrices over the complex numbers and the corresponding graded polynomial identities. Given a group and a fine grading on , we show that the ideal of graded identities is generated by a special type of identity, and, as a consequence, we solve the corresponding Specht problem for this case. Next we construct a universal algebra (depending on the group and the grading) in two different ways: one by means of polynomial identities and the other one by means of a generic twococycle (this parallels the classical constructions in the nongraded case). We show that a suitable central localization of is Azumaya over its center and moreover, its homomorphic images are precisely the graded forms of . Finally, we consider the ring of central quotients of which is a central simple algebra over the field of quotients of the center of . Using earlier results of the authors we show that this is a division algebra if and only if the group is one of a very explicit (and short) list of nilpotent groups. It follows that for groups not on this list, one can find a nonidentity graded polynomial such that its power is a graded identity. We illustrate this phenomenon with an explicit example.
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Polynomial identities and noncommutative versal torsors, arXiv:0708.4108, Adv. Math. (2008), doi:10.1016/j.aim.2008.03.014.
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On the universal graded central simple algebra, Actas del XVI Coloquio Latinoamericano de Álgebra (Colonia, Uruguay, 2005), 115130, Revista Matemática Iberoamericana, Madrid, 2007.
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 E. Aljadeff, J. Sonn,
Projective Schur algebras of nilpotent type are Brauer equivalent to radical algebras, J. Algebra 220 (1999) 401414. MR 1717348 (2000i:16035)
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 Yu. Bahturin, V. Drensky,
Graded polynomial identities of matrices, Linear Algebra Appl. 357 (2002), 1534. MR 1935223 (2003k:16034)
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Additional Information
Eli Aljadeff
Affiliation:
Department of Mathematics, TechnionIsrael Institute of Technology, Haifa 32000, Israel
Email:
aljadeff@tx.technion.ac.il
Darrell Haile
Affiliation:
Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Email:
haile@indiana.edu
Michael Natapov
Affiliation:
Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Address at time of publication:
Department of Mathematics, TechnionIsrael Institute of Technology, Haifa 32000, Israel
Email:
mnatapov@indiana.edu, natapov@tx.technion.ac.il
DOI:
http://dx.doi.org/10.1090/S0002994710048117
Received by editor(s):
October 26, 2007
Received by editor(s) in revised form:
April 22, 2008
Published electronically:
January 7, 2010
Article copyright:
© Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
